Q1. a) Solve i. \( x^{2}+y^{A^{2}}=4 \) and \( y=2 \) ii. \( x^{2}+y^{7}=4 \) and \( y=x+1 \) b) what does your answer in (a) represent?

Alright, I need to solve the given system of equations and then interpret the results. Let's tackle part (a) first. **Part (a):** **i. \( x^{2} + y^{A^{2}} = 4 \) and \( y = 2 \)** First, I'll substitute \( y = 2 \) into the first equation to find the value of \( x \). So, the equation becomes: \[ x^{2} + (2)^{A^{2}} = 4 \] Now, I need to solve for \( x \). However, there's a variable \( A \) in the exponent of \( y \), which complicates things. Since \( A \) is not defined, I'll assume it's a constant for the sake of solving. Let's denote \( A^{2} = k \), where \( k \) is a constant. So, the equation simplifies to: \[ x^{2} + 2^{k} = 4 \] Subtracting \( 2^{k} \) from both sides: \[ x^{2} = 4 - 2^{k} \] Taking the square root of both sides: \[ x = \sqrt{4 - 2^{k}} \] But since \( x \) can be positive or negative, the solutions are: \[ x = \sqrt{4 - 2^{k}} \quad \text{and} \quad x = -\sqrt{4 - 2^{k}} \] However, without a specific value for \( k \) (which depends on \( A \)), I can't provide numerical solutions. If \( A \) is given, I can substitute it back to find the exact values of \( x \). **ii. \( x^{2} + y^{7} = 4 \) and \( y = x + 1 \)** Again, I'll substitute \( y = x + 1 \) into the first equation to eliminate one variable. Substituting: \[ x^{2} + (x + 1)^{7} = 4 \] Now, I need to solve for \( x \). Expanding \( (x + 1)^{7} \) using the binomial theorem: \[ (x + 1)^{7} = x^{7} + 7x^{6} + 21x^{5} + 35x^{4} + 35x^{3} + 21x^{2} + 7x + 1 \] So, the equation becomes: \[ x^{2} + x^{7} + 7x^{6} + 21x^{5} + 35x^{4} + 35x^{3} + 21x^{2} + 7x + 1 = 4 \] Bringing all terms to one side: \[ x^{7} + 7x^{6} + 21x^{5} + 35x^{4} + 35x^{3} + 22x^{2} + 7x - 3 = 0 \] This is a 7th-degree polynomial equation, which is quite complex to solve analytically. Typically, such equations are solved numerically using methods like Newton-Raphson or graphing techniques to approximate the roots. Without specific tools or further information, I can't provide exact solutions here. **Part (b): What does your answer in (a) represent?** The solutions to the system of equations in part (a) represent the points of intersection between the two curves defined by the equations. Specifically: - In part (a)i, the equation \( x^{2} + y^{A^{2}} = 4 \) represents a hyperbola-like curve, and \( y = 2 \) is a horizontal line. The solutions \( x = \sqrt{4 - 2^{k}} \) and \( x = -\sqrt{4 - 2^{k}} \) indicate the points where the hyperbola intersects the horizontal line \( y = 2 \). - In part (a)ii, the equation \( x^{2} + y^{7} = 4 \) represents a curve that is more complex due to the exponent 7, and \( y = x + 1 \) is a straight line with a slope of 1 and a y-intercept of 1. The solutions to this system would represent the points where the 7th-degree curve intersects the straight line \( y = x + 1 \). These intersection points are significant in various applications, such as finding equilibrium points in physics, determining optimal solutions in economics, or solving geometric problems where two shapes intersect.